Variations on a theme of Frobenius about almost commuting unitary matrices

Let A and B be two rotations in Rn so that B does not rotate any vector by an angle of π2 radians. F.G. Frobenius proved that if A commutes with the commutator [A, B] = A−1B−1AB, then A and B are commuting rotations. We prove a generalisation for almost commuting unitary matrices, giving explicit estimations of the error terms, which can be useful in numerical matrix computations.